GeoGebra Classroom

Locus of the centers of all circles tangent to two circles

Given are two circles with centers and . In this applet we construct all circles tangent to the given circles and study the locus of their centers.
• Drag the orange point along the circle or click the animation button to see one set of tangent circles.
• Drag one of the blue circles to a different location – touching the other blue circle, intersecting the circle, in the interior of the larger circle.
• Prove that the locus of the centers of the red circles is a hyperbola with foci at the centers of the circles and distance between the vertices equal to the difference of the radii of the circles. When is the locus an ellipse?
• Click on “Show Circle 2” checkbox to show another tangent circle.
• Click on “Show Locus 2”.
• Drag one of the blue circles to a different location and see how the locus is changing.
• Prove that the locus of the centers of the green circles is a hyperbola with foci at the centers of the given circles and distance between the vertices equal to the sum of the radii of the given circles. When is the locus an ellipse?